TIMEOUT
The TRS could not be proven terminating. The proof attempt took 60001 ms.
The following DP Processors were used
Problem 1 was processed with processor DependencyGraph (36ms).
| – Problem 2 was processed with processor SubtermCriterion (2ms).
| – Problem 3 remains open; application of the following processors failed [SubtermCriterion (0ms), DependencyGraph (1ms), PolynomialLinearRange4iUR (392ms), DependencyGraph (1ms), PolynomialLinearRange8NegiUR (3279ms), DependencyGraph (1ms), ReductionPairSAT (345ms), DependencyGraph (0ms), SizeChangePrinciple (143ms), ForwardNarrowing (1ms), BackwardInstantiation (1ms), ForwardInstantiation (3ms), Propagation (0ms)].
The following open problems remain:
Open Dependency Pair Problem 3
Dependency Pairs
| cond#(true, x) | → | cond#(and(even(x), gr(x, 0)), p(x)) |
Rewrite Rules
| cond(true, x) | → | cond(and(even(x), gr(x, 0)), p(x)) | | and(x, false) | → | false |
| and(false, x) | → | false | | and(true, true) | → | true |
| even(0) | → | true | | even(s(0)) | → | false |
| even(s(s(x))) | → | even(x) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: 0, s, p, false, true, even, gr, cond, y, and
Problem 1: DependencyGraph
Dependency Pair Problem
Dependency Pairs
| cond#(true, x) | → | even#(x) | | cond#(true, x) | → | gr#(x, 0) |
| cond#(true, x) | → | p#(x) | | even#(s(s(x))) | → | even#(x) |
| cond#(true, x) | → | and#(even(x), gr(x, 0)) | | cond#(true, x) | → | cond#(and(even(x), gr(x, 0)), p(x)) |
| gr#(s(x), s(y)) | → | gr#(x, y) |
Rewrite Rules
| cond(true, x) | → | cond(and(even(x), gr(x, 0)), p(x)) | | and(x, false) | → | false |
| and(false, x) | → | false | | and(true, true) | → | true |
| even(0) | → | true | | even(s(0)) | → | false |
| even(s(s(x))) | → | even(x) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: 0, s, p, true, false, even, gr, cond, y, and
Strategy
The following SCCs where found
| even#(s(s(x))) → even#(x) |
| cond#(true, x) → cond#(and(even(x), gr(x, 0)), p(x)) |
Problem 2: SubtermCriterion
Dependency Pair Problem
Dependency Pairs
| even#(s(s(x))) | → | even#(x) |
Rewrite Rules
| cond(true, x) | → | cond(and(even(x), gr(x, 0)), p(x)) | | and(x, false) | → | false |
| and(false, x) | → | false | | and(true, true) | → | true |
| even(0) | → | true | | even(s(0)) | → | false |
| even(s(s(x))) | → | even(x) | | gr(0, x) | → | false |
| gr(s(x), 0) | → | true | | gr(s(x), s(y)) | → | gr(x, y) |
| p(0) | → | 0 | | p(s(x)) | → | x |
Original Signature
Termination of terms over the following signature is verified: 0, s, p, true, false, even, gr, cond, y, and
Strategy
Projection
The following projection was used:
Thus, the following dependency pairs are removed:
| even#(s(s(x))) | → | even#(x) |